Fix the circle x2 + (y − a/2)2 = (a/2)2 and the line y = a. For each θ in the interval (0,...

Fix the circle x2 + (y − a/2)2 = (a/2)2 and the line y = a. For each θ in the interval (0, π), construct a line through the origin that forms angle θ with the positive x-axis. Let B be the point at which it intersects the circle, A the point at which it intersects the line, and P the point below A such that ΔPAB is a right triangle. Find a parametrization for the path followed by P as θ ranges over (0, π).

In the United States and England, this curve is called the Witch of Agnesi. The reason for this curious appellation is a mistake made by the Cambridge Lucasian Professor of Mathematics, John Colson, in about 1801 in translating a calculus textbook from Italian to English. In the book Instituzioni analitiche ad uso delta gioventu italiana, published in 1748, its author Maria Gaetana Agnesi described this “versed sine curve” and used the Latin word averisera for it. Colson evidently read the word as the Italian aversiera, which means “witch” and the rest is, well, history. See [6] for more details.